2 Answers
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An apostle is one sent from God, a disciple is one who studied under God.

Decent (albeit imperfect) analogues in the terminology would be Apostle is to Missionary as Disciple is to Student.

The 12 disciples included all of the men who studied under Jesus - including Judas.

Once Jesus died and rose again, he sent the remaining 11 of them out (obviously not Judas, since he was dead!) to preach the Good News to the ends of the earth (Matt 28:19-20), and later called Paul to do the same.

Paul, however, did not know Jesus until his conversion on the road to Damascus (Acts 9), well after Jesus' teaching ministry was done. Thus, Paul was sent out (apostolos), but was never discipled directly by Jesus.

Paul was an Apostle because that's what he said he was, and others said he was (see Acts 14:14) "But when the apostles Barnabas and Paul heard of this, they tore their clothes and rushed out into the crowd, shouting:".
Paul was not one of the 12 original discipels, he was converted on the road to Damascus when He saw Jesus, who he had been previously persecuting. So Paul really couldn't be called a discipel .
Most of Paul's letters started with him asserting his Apostleship. See 1 Tim 1:1 "Paul, an apostle of Christ Jesus by the command of God our Savior and of Christ Jesus our hope,". And Paul tells the Corinthians in 1 Corinthians 9:2 that they are the seal of his apostleship, weather or not anyone else recognizes him.
Peter even includes Pauls writings with "the rest of scripture" in 2nd Peter 3. further legitimizing his Apostolic authority, and not only that but when Peter himself was in error Paul corrected him(see Galatians 2:11-14)